April 25, 2012

## A bit of History about Pi

Mainstream historians believe that ancient Egyptians had no concept of π

As early as the 19th century BCE, Babylonian mathematicians were using π ≈ 25/8, which is about 0.5 percent below the exact value

The Indian astronomer Yajnavalkya gave astronomical calculations in the Shatapatha Brahmana (c. 9th century BCE) that led to a fractional approximation of π ≈ 339/108 (which equals 3.13888…, which is correct to two decimal places when rounded, or 0.09 percent below the exact value).

Recall an n-gon is a polygon with n sides.

The first recorded algorithm for rigorously calculating the value of π was a geometrical approach utilizing polygons which was used around 250 BC by Greek mathematician Archimedes.

Archimedes of Syracuse

(Greek: Ἀρχιμήδης; c. 287 BC – c. 212 BC)

This polygonal algorithm remained the primary approach for computing π for over 1,000 years. Archimedes computed upper and lower bounds of π by drawing regular polygons inside and outside a circle, and calculating the perimeters of the outer and inner polygons as you can see below.

By using the equivalent of 96-sided polygons, he proved that :

223/71 < π < 22/7.

Archimedes’ upper bound of 22/7 may have led to widespread belief that π was equal to 22/7. Around 150 AD, Greek-Roman scientist Ptolemy, in his book Almagest, gave a value for π of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga. Mathematicians using polygonal algorithms reached 39 digits of π in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.

(From Wikipedia, several articles)

Nowadays we know hundred of thousands of decimal figures of PI, but using computers!

## A  joke

Time to relax

See you tomorrow!

## Lesson 8: A Tale about Thales

April 5, 2012

 Born Approximately 624 BC,Miletus,Asia Minor. (Now Balat,Turkey) Died Approximately 547 BC

Thales, an engineer by trade, was the first of the Seven Sages, or wise men of Ancient Greece. Thales is known as the first Greek philosopher, mathematician and scientist. He founded the geometry of lines, so is given credit for introducing abstract geometry.

He was the founder of the Ionian school of philosophy in Miletus, and the teacher of Anaximander. During Thales’ time, Miletus was an important Greek metropolis in Asia Minor, known for scholarship. Several schools were founded in Miletus, attracting scientists, philosophers, architects and geographers.

Thales is credited with introducing the concepts of logical proof for abstract propositions.

Thales went to Egypt and studied with the priests, where he learned of mathematical innovations and brought this knowledge back to Greece. Thales also did geometrical research and, using triangles, applied his understanding of geometry to calculate the distance from shore of ships at sea. This was particularly important to the Greeks, whether the ships were coming to trade or to do battle.

But this is too serious, here it is your tale:

About 600 BC Thales was visiting Egypt. The Pyramids were built 2000 years ago. While Thales was in Egypt, he was called by the Pharaoh who knew his fame of wise man. He proposed an old problem:

-“Find out the exact height of the Great Pyramid and I will cover you with gold”

Thales went to the Great Pyramid and leaned on his  stick and waited. When the stick casted a shadow equal to its length he said to a servant:

– “Run quickly and measure the shadow of the Great Pyramid because at this moment it is as long as the Pyramid”.

This way the Pharaoh made him part of Pharaoh court of wise men and covered him with gold.

Thales learned about the Egyptian rope-pullers and their methods of surveying land for the Pharaoh using stakes and ropes. Property boundaries had to be re-established each year after the Nile flooded. After Thales returned to Greece about 585 BC with notes about what he had learned, and Greek mathematicians translated the rope-and-stake methods of the rope pullers into a system of points, lines and arcs. They also took geometry from the fields to the page by employing two drawing tools, the straightedge for straight lines and the compass for arcs

There are many recorded tales about Thales, some complimentary and others critical:

• Herodotus noted that Thales predicted the solar eclipse of 585 BC, a notable advancement for Greek science.
• Aristotle reported that Thales used his skills at recognizing weather patterns to predict that the next season’s olive crop would be bountiful. He purchased all the olive presses in the area, and made a fortune when the prediction came true!
• Plato told a story of Thales gazing at the night sky, not watching where he walked, and so fell into a ditch. The servant girl who came to help him up then said to him “How do you expect to understand what is going on up in the sky if you do not even see what is at your feet?”. Perhaps this is the first absent-minded professor joke in the West!
Quotations attributed to Thales
• “A multitude of words is no proof of a prudent mind.”
• “Hope is the poor man’s bread.”
• “The past is certain, the future obscure.”
• “Nothing is more active than thought, for it travels over the universe, and nothing is stronger than necessity for all must submit to it.”
• “Know thyself.”
The historical notes are an abstract from:
http://www.mathopenref.com/thales.html by Charlene Douglass, California, 2006.

## Lesson 7: Who Was Diophantus? Diophantus’s Riddle (IV)

February 25, 2012

### Diophantus of Alexandria

Diophantus is often known as the ‘father of algebra’,but there is no doubt that many of the methods for solving linear and quadratic equations go back to Babylonian mathematics. Nevertheless, his remarkable, collection of problems is a singular achievement that was not fully appreciated and further developed until much later.

He is best known for his Arithmetica, a work on the solution of algebraic equations and on the theory of numbers. The Arithmetica is a collection of 130 problems giving numerical solutions of determinate equations (those with a unique solution), and indeterminate equations. The method for solving the latter is now known as Diophantine analysis. Only six of the original 13 books were thought to have survived and it was also thought that the others must have been lost quite soon after they were written. Diophantus was the first Greek mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficients and solutions.

Diophantus did not use sophisticated algebraic notation, he did introduce an algebraic symbolism that used an abbreviation for the unknown and for the powers of the unknown.

However, essentially nothing is known of his life and there has been much debate regarding the date at which he lived.

You can read a good biography on

where I found this information. There are many biographies of famous mathematicians.

### Diophantus Riddle

Much of our knowledge of the life of Diophantus is derived from a 5th century Greek anthology of number games and strategy puzzles. One of the problems (sometimes called his epitaph) states:

‘Here lies Diophantus,’ the wonder behold.

Through art algebraic, the stone tells how old:

‘God gave him his boyhood one-sixth of his life,

One twelfth more as youth while whiskers grew rife;

And then yet one-seventh ere marriage begun;

In five years there came a bouncing new son.

Alas, the dear child of master and sage

After attaining half the measure of his father’s life chill fate took him.

After consoling his fate by the science of numbers for four years, he ended his life.’

This puzzle implies that Diophantus lived …..Could  you tell it to us?

### You will publish a post with your solution on this blog and there is an interesting reward for you!

However, the accuracy of the information cannot be independently confirmed.

## Lesson 4: Numeric Proportionality. History of the Rule of Three (III)

March 31, 2011

### The Rule of Three

The rule of three was a name given in earlier days to an algorithm for solving proportions. The method required setting up the problem so that the unkown quantity is always last “extreme” in the proportionality. In  http://www.pballew.net/arithm18.html you can see an image that shows the rules as given by a 1827 arithmetic.

This rule is covered in almost all the arithmetics up to the beginning of the 20th century.

The rule of three was such a common part of arithmetic education that it found its way into common expressions. In his autobiography, Lincoln writes that he that he learned to “read, write, and cipher to the rule of 3.” A poem often used in student copy books was:

Multiplication is vexation;